L(s) = 1 | + (−1.93 + 1.11i)5-s + (−2.26 + 3.92i)7-s + (12.9 + 7.46i)11-s + (−7.82 − 13.5i)13-s − 7.53i·17-s + 9.77·19-s + (−9.38 + 5.41i)23-s + (2.5 − 4.33i)25-s + (3.94 + 2.27i)29-s + (24.4 + 42.4i)31-s − 10.1i·35-s + 21.4·37-s + (45.6 − 26.3i)41-s + (−20.5 + 35.6i)43-s + (−32.2 − 18.6i)47-s + ⋯ |
L(s) = 1 | + (−0.387 + 0.223i)5-s + (−0.323 + 0.560i)7-s + (1.17 + 0.678i)11-s + (−0.602 − 1.04i)13-s − 0.443i·17-s + 0.514·19-s + (−0.408 + 0.235i)23-s + (0.100 − 0.173i)25-s + (0.136 + 0.0785i)29-s + (0.789 + 1.36i)31-s − 0.289i·35-s + 0.580·37-s + (1.11 − 0.642i)41-s + (−0.478 + 0.829i)43-s + (−0.685 − 0.395i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.409878747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409878747\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
good | 7 | \( 1 + (2.26 - 3.92i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-12.9 - 7.46i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (7.82 + 13.5i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 7.53iT - 289T^{2} \) |
| 19 | \( 1 - 9.77T + 361T^{2} \) |
| 23 | \( 1 + (9.38 - 5.41i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-3.94 - 2.27i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-24.4 - 42.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-45.6 + 26.3i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (20.5 - 35.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.2 + 18.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 81.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (70.3 - 40.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.5 - 42.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-23.2 - 40.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 17.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 101.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (28.7 - 49.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-75.2 - 43.4i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (90.4 - 156. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.505156427635028066952203487260, −8.656470004225360780615962650508, −7.76612320687610132540456975677, −7.04461926961041687046670034789, −6.25913803242374575189816762267, −5.30160261267680463820668943071, −4.42377608859260268311970779587, −3.35983029109675629571255952927, −2.53814067159643402082007436164, −1.11241837567071996716151303554,
0.43593448156658952356456738518, 1.63174080020162503762083608186, 3.03855618355321968351294598461, 4.07543057639064987942228575626, 4.52920159998597149152748515008, 5.95203433014204066359748933516, 6.52751727174616221570814920126, 7.43132798110101744187822598273, 8.160314949456473924578921178896, 9.155570592821691686521623984023